What is Hermitian: Definition and 349 Discussions

I'm trying to derive something which shouldn't be too complicated, but I get different results when doing things symbolically and with actual operators and wave functions. Some help would be appreciated. For the hydrogenic atom, I need to calculate ##\langle \hat{H}\hat{V} \rangle## and...

I am looking at the derivation of the Heisenberg Uncertainty Principle presented here: http://socrates.berkeley.edu/~jemoore/p137a/uncertaintynotes.pdf and am confused about line (21)... I do not understand why AB and BA are complex conjugates of each other... (I'm still in high school so I...

Homework Statement I have a hermitian Operator A and a quantum state |Psi>=a|1>+b|2> (so we're an in a two-dim. Hilbert space) In generally, {|1>,|2>} is not the eigenbasis of the operator A. I shall now show that the Eigenvaluse of A are the maximal (minimal) expection values <Psi|A|Psi>.The...

Homework Statement Hi, I'm doing a Quantum chemistry and one of my question is to determine if is hermitian or not? I am learning and new to this subject... Cant figure out how to do this question at all. Please helppp! ^Q= i/x^2 d/dx is hermitian or not? Homework Equations The Attempt at a...

Homework Statement Find the eigenvalues and normalized eigenfuctions of the following Hermitian operator \hat{F}=\alpha\hat{p}+\beta\hat{x} Homework Equations In general: ##\hat{Q}\psi_i = q_i\psi_i## The Attempt at a Solution I'm a little confused here, so for example I don't know...

I was wondering what the difference is between the two. Would be nice if someone could explain the difference in simple terms, because it appears to be essential to my quantum mechanics course.

Definition/Summary The Hermitian transpose or Hermitian conjugate (or conjugate transpose) M^{\dagger} of a matrix M is the complex conjugate of its transpose M^T. A matrix is Hermitian if it is its own Hermitian transpose: M^{\dagger}\ =\ M. An operator A is Hermitian (or self-adjoint)...

I have an operator which isnot Hermitian is there any way to make it hermitian ?

Hi I have been looking at the solutions to a past exam question. The question gives the annihilation operator for the harmonic oscillator as a= x + ip ( I have left out the constants ). The question then asks to calculate the Hermitian conjugate a(dagger). I thought to find the Hermitian...

Hi. In 2-fold degenerate perturbation theory we can find appropiate "unperturbate" wavefunctions by looking for simultaneous eigenvectors (with different eigenvalues) of and H° and another Hermitian operator A that conmutes with H° and H'. Suppose we have the eingenvalues of H° are ##E_n =...

I am a QM beginner so go easy on me. I have just noticed something. Let $$\hat{O}$$ be an hermitian operator. Then $$\left( \hat { O } \right) ^{ \dagger }\neq \hat { O } $$ when it is by itself. For example $$\left( \hat { p } \right) ^{ \dagger }=i\hbar \frac { \partial }{ \partial x...

Homework Statement If Hamiltonian ##\hat{H}## is hermitian show that operator ##\hat{\rho}=\frac{e^{-\beta\hat{H}}}{Tr(e^{-\beta\hat{H}})}## is statistical.Homework Equations In order to be statistical operator ##\hat{\rho}## must be hermitian and must have trace equal ##1##...

hello i have to proof that Px (linear momentum operator ) is hermitian or not i have added my solution in attachments please look at my solution and tell me if its correct thank you all

Hello, If I have a Hermitian matrix A, can I write it as: \mathbf{A}=\mathbf{A}^{1/2}\mathbf{A}^{H/2} where superscript H denotes hermitian operation? Thanks

Hi, i was wondering how the following expression can be decomposed: Let A=B°C, where B, C are rectangular random matrices and (°) denotes Hadamard product sign. Also, let (.) (.)H denote Hermitian transposition. Then, AH *A how can be decomposed in terms of B and C ?? For example, AH...

Is there any way to write the Dirac lagrangian to have symmetric derivatives (acting on both sides)? Of course someone can do that by trying to make the Lagrangian completely hermitian by adding the hermitian conjugate, and he'll get the same equations of motion (a 1/2 must exist in that...

Homework Statement Spin Operator S has eigenvectors |R> and |L>, S|R> = |R> S|L> =-|L> eigenvectors are orthonormal Homework Equations Operator A is Hermitian if <ψ|A|Θ> = <Θ|A|ψ>* The Attempt at a Solution <ψ|S|L> = <L|S|ψ>* // Has to be true if S is Hermitian LHS...

In the standard QFT textbook, the Hermitian conjugate of a Dirac field bilinear \bar\psi_1\gamma^\mu \psi_2 is \bar\psi_2\gamma^\mu \psi_1. Here is the question, why there is not an extra minus sign coming from the anti-symmetry of fermion fields?

I'm currently reading the book Introductory Quantum Mechanics by Richard Liboff 4th edition. I'm reading one of the proofs and I don't understand what is happening in one of the steps. The problem is trying to find the Hermitian adjoint of the operator \hat{D}=\partial/\partialx defined in...

We know that all eigenvalues of a Hermitian matrix are real. How to explain this from the physics point of view?

A Hermitian operator A is defined by A=A(dagger) which is the transpose and complex conjugate of A. In 1-D the momentum operator is -i(h bar)d/dx. How can this be Hermitian as the conjugate has the opposite sign ? Thanks

I can't figure this one out given that the coordinate operator is continuous, it's hard to imagine "matrix elements". But presumably since the coordinates of the system (1d free particle) are always real valued, would this make the coordinate operator Hermitian?

This is certainly an elementary question, so I would be all the more grateful for the answer. Given: A and B are two Hermitian operators and v is a vector in C. If <v|AB|v>=x+iy (for x and y real), then <v|BA|v> = x-iy. Why?

Homework Statement Linear combination is \hat{A} + i\hat{B}. It's given that it is not Hermitian already. Homework Equations ∫ψi * \hat{Ω} ψj = (∫ψj * \hat{Ω} ψi)* The Attempt at a Solution ∫ψi * (\hat{A} + i\hat{B}) ψj = (∫ψj * (\hat{A} + i\hat{B}) ψi)* I chose to work with...

Hi, I have a 3 by 3 hermitian matrix K that I need to diagonalise. More accurately, I am searching for a unitary matrix S such that S^{\dagger} K S is diagonal. The problem is that K is very complicated and the expression for S in mathematica takes up quiiiiiet a lot of space. Is it...

Why generator matrices of a compact Lie algebra are Hermitian?I know that generators of adjoint representation are Hermitian,but how about the general representaion of Lie groups?

Homework Statement Using Dirac Notation prove for the Hermitian operator B acting on a state vector |ψ>, which represents a bound particle in a 1-d potential well - that the expectation value is <C^2> = <Cψ|Cψ>. Include each step in your reasoning. Finally use the result to show the...

Hi, I know this question may seem a little trivial, but is there any real difference between \left (\partial_{\mu} \phi \right)^{\dagger} and \partial_{\mu} \phi^{\dagger} and if so, could someone provide an explanation? Many thanks. (Sorry if this isn't quite in the right...

As we know, all operators representing observables are Hermitian. In my undersatanding, this statement means that all operators representing observables are Hermitian if the system can be described by a wavefunction or a vector in L2. For example, the momentum operator p is Herminitian...

Particle Number Operator (Hermitian??) Hey guys, I'm studying the quantic harmonic oscillator and I'm using "Cohen-Tannoudji Quantum Mechanics Volume 1". At some point he introduced the particle number operator, N, such that: N=a+.a , where a+ is the conjugate operator of a. The...

Hey guys, So this question is sort of a fundamental one but I'm a bit confused for some reason. Basically, say I have a Hermitian operator \hat{A}. If I have a system that is prepared in an eigenstate of \hat{A}, that basically means that \hat{A}\psi = \lambda \psi, where \lambda is real...

Hi everyone, :) Here's a problem I encountered. I think there's a mistake in this problem. Problem: Let \(f:\,V\times V\rightarrow\mathbb{C}\) be a Hermitian function (a Bilinear Hermitian map), \(q:\, V\rightarrow\mathbb{C}\) be given by \(q(v)=f(v,\,v)\). Prove that following...

Let's say we have operator X that is Hermitian and we have operator P that is Hermitian. Is the following true: [X,P]=ihbar This is the commutator of X and P. This particular result is known as the canonical commutation relation. Expanding: [X,P]=XP-PX=ihbar This result indicates that...

Prove the equation A\left|\psi\right\rangle = \left\langle A\right\rangle\left|\psi\right\rangle + \Delta A\left|\psi\bot\right\rangle where A is a Hermitian operator and \left\langle\psi |\psi\bot\right\rangle = 0 \left\langle A\right\rangle = The expectation value of A. \Delta A...

Two hermitian commutator anticommute: {A,B}=AB+BA=0.Is it possible to have a simultaneous eigenket of A and B?illustrate... Thank you in advance

Homework Statement [A,B] = C and operators A,B,C are all hermitian show that C=0 Homework Equations The Attempt at a Solution Since it is given that all operators are hermitian I know that A=A' B=B' and C=C' so i expanded it out to AB-BA=C A'B'-B'A'=C (BA)' - (AB)'=C...

The problem asks to show that the kinetic energy operator is Hermitian. The operator is given as T= -h^2/2mΔ but I know I can also write it as p^2/2m which would be (- ih∇)(-ih∇). My main question is if I can prove this in 1-D so that T=(-h^2/2m)d^2/dx^2 does that generalize to...

I am trying to prove that for any two vectors x,y in ##ℂ^{n}## the product ## \langle x,y \rangle = xAy^{*} ## is an inner product where ##A## is an ##n \times n## Hermitian matrix. This is actually a generalized problem I created out of a simpler textbook problem so I am not even sure if this...

Find its eigen values. Is this operator linear?

¿ Is it the same self-adjoint operator that hermitian operator If it is not the same, what is the difference? And an observable is an operator whose eigenvectors form basis in the Hilbert space, and it is hermitian, or self-adjoint? I always considered both terms like sinonynms, in the...

The regular spin orbit Hamiltonian is H_{SO} = \frac{q\hbar}{4 m^2 c^2}\sigma\cdot(\textbf{E}\times \textbf{p}) If I consider a 2D system where E = E(x,y) and p is treated as an operator, i.e. \hat{p} = \hat{i}p_x + \hat{j}p_y then, clearly E and p do not commute, so this doesn't look like...

Is the following a theorem? yes or no If A and B are non-commuting Hermitian operators (or matrices), there does not exist Hermitian operators C and D such that AB-BA = CD. (Or, as special case, ...there does not exist a Hermitian operator C s.t. C= AB-BA) Thanks

Let H and K be hermitian operators on vector space U. Show that operator HK is hermitian if and only if HK=KH. I tried some things but I don't know if it is ok. Can somebody please check? I got a hint on this forum that statements type "if only if" require proof in both directions, so here...

Homework Statement If B is Hermitian, show that BN and the real, smooth function f(B) is as well. Homework Equations The operator B is Hermitian if \int { { f }^{ * }(x)Bg(x)dx= } { \left[ \int { { g }^{ * }(x)Bf(x) } \right] }^{ * } The Attempt at a Solution Below is my...

Homework Statement Find the following hermitian conjugates and show if they are hermitian operators: i) xp ii) [x , p] iii) xp + px Where x is the position operator and p is the momentum operator. Homework Equations <f|Qg> = <Q^{t}f|g> Q = Q^{t} Hermitian operator p =...

Hi there, This should be very simple... If I have a state <1|AB|2> where A and B are Hermitian operators, can I rewrite this as <2|BA|1> ? That would be, taking the complex conjugate of the matrix element and saying that A*=A and B*=B. Thank you!

Greetings, My task is to prove that the angular momentum operator is hermitian. I started out as follows: \vec{L}=\vec{r}\times\vec{p} Where the above quantities are vector operators. Taking the hermitian conjugate yields \vec{L''}=\vec{p''}\times\vec{r''} Here I have used double...

Hi, Does anyone know how to prove that two commutative Hermitian matrices can always have the same set of eigenvectors? i.e. AB - BA=0 A and B are both Hermitian matrices, how to prove A and B have the same set of eigenvectors? Thanks!

Hello everyone, I found that you're actively discussing math problems here and thought to share my problem with you. [Givens:] I have a specially structured complex-valued n \times n matrix, that has only three non-zero constant diagonals (the main diagonal, the j^{th} subdiagonal and the...

Hey, I have the following question on Hermitian operators Initially I thought this expectation value would have to be zero as the eigenvectors are mutually orthogonal due to Hermitian Operator and so provided the eigenvectors are distinct then the expectation would be zero... Though...